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Definition df-ms 12523
Description: Define the (proper) class of metric spaces. (Contributed by NM, 27-Aug-2006.)
Assertion
Ref Expression
df-ms MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}

Detailed syntax breakdown of Definition df-ms
StepHypRef Expression
1 cms 12520 . 2 class MetSp
2 vf . . . . . . 7 setvar 𝑓
32cv 1330 . . . . . 6 class 𝑓
4 cds 12044 . . . . . 6 class dist
53, 4cfv 5123 . . . . 5 class (dist‘𝑓)
6 cbs 11973 . . . . . . 7 class Base
73, 6cfv 5123 . . . . . 6 class (Base‘𝑓)
87, 7cxp 4537 . . . . 5 class ((Base‘𝑓) × (Base‘𝑓))
95, 8cres 4541 . . . 4 class ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))
10 cmet 12164 . . . . 5 class Met
117, 10cfv 5123 . . . 4 class (Met‘(Base‘𝑓))
129, 11wcel 1480 . . 3 wff ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))
13 cxms 12519 . . 3 class ∞MetSp
1412, 2, 13crab 2420 . 2 class {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
151, 14wceq 1331 1 wff MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
Colors of variables: wff set class
This definition is referenced by:  isms  12636
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